In his seminal article [1]}, Robert Bryant introduced the following formalism for studying the Willmore immersions of a surface via its conformal Gauss map. Let \(\vec{\Phi }:\Sigma \rightarrow \mathbb {R}^3\)  be an immersion with unit normal vector \(\vec{n}\) . Let \(\mathbb {S}^{4,1}\)  be the deSitter space. The conformal Gauss map \(\vec{Y}:\Sigma \rightarrow \mathbb {S}^{4,1}\)  is defined via
\(\vec{Y}:= H \vec{X}+ \vec{N} \nonumber \) 
